Cant select/move multiple keyframes even with B select?
The dopesheet behaves similiar to the 3D view.

The right mouse is used for selecting objects or keys. Incidentically, the RMB followed by a movement before releasing the mousebutton selects an object/key and immediately enters the translation operation.

But since you only want to do a translation, you have to initiate it with a shortcut as intended. The correct way is to do a box selection and then...


make sure the cursor is hovering in the dopesheet (so it will receive the keys pressed and not some other window; Blender is context sensitive)
press G (grab) to enter a translation operation (G key down and up)
move the mouse, (moving the keys)
press the left mouse button to confirm the operation and exit the translation.


OR

4a. press the right mouse button to escape the operation. The keys will snap back into place.